GCSE Revision Notes
Mathematics –
‘Quadratic Formula’
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Quadratic Formula
A quadratic equation has standard form
The numbers a, b and c are the coefficients of the equation, while x is said to be a
variable.
Note that in the standard form all non-zero terms appear in one side, and the other side
is always zero.
The solutions of an equation are the values of the variable x for which the equation is
true.
Therefore, to find the solutions of a quadratic equation we must find the values for
which
the expression
is equal to 0.
If solutions of the equations exist we usually find them by factorization. However, when
factorization is not an option we can solve the equation by using the quadratic formula.
√
The quantity
is called the discriminant of the quadratic equation, and is usually
represented by the symbol Δ (greek uppercase delta).
We can rewrite the above as:
√
where
The discriminant Δ can be positive, negative or zero.
 If Δ > 0
There are two distinct roots (solutions)
 If Δ = 0
There is 1 root
 If Δ < 0
There are no real roots
Examples

Question 4.1
In the equation
Indicate a, b and c.
Determine Δ and indicate whether the equation has 1, 2 or no solutions.
Answer
We have:
a=1
b=5
c=3
And the determinant is
The determinant is positive, therefore there are two distinct roots.

Question 4.2
In the equation
Indicate a, b and c.
Determine Δ and indicate whether the equation has 1, 2 or no solutions.
Answer
We have:
a = 12
b=-4
c=2
And the determinant is
(
)
The determinant is negative, therefore there are no real roots.

Question 4.3
In the equation
Indicate a, b and c.
Determine Δ and indicate whether the equation has 1, 2 or no solutions.
Answer
We have:
a=2
b=-4
c=2
And the determinant is
(
)
The determinant is zero, therefore there is 1 real root.

Question 4.4
In the equation
Indicate a, b and c.
Determine Δ and indicate whether the equation has 1, 2 or no solutions.
Answer
We have:
a=-3
b=3
c=
And the determinant is
(
) (
)
The determinant is zero, therefore there is 1 real root.

Question 4.5
In the equation
Indicate a, b and c.
Determine Δ and indicate whether the equation has 1, 2 or no solutions.
Answer
Note that the term containing x is not present.
To say that there is no x is the same of saying that its coefficient is zero (0x).
We have:
a = 26
b=0
c=
And the determinant is
(
(
) (
)
)
The determinant is positive, therefore there are two distinct roots.

Question 4.6
In the equation
Indicate a, b and c.
Determine Δ and indicate whether the equation has 1, 2 or no solutions.
Answer
Here the constant term c does not appear. We write that as c = 0 .
We have:
a=-1
b=-5
c=
And the determinant is
(
)
(
) ( )
The determinant is positive, therefore there are two distinct roots.
We now use the quadratic formula to find roots (if any) of quadratic equations

Question 4.7
Find the roots (if any) of the equation
Answer
Here we have:
a=
b=-5
c=
Calculate the determinant
(
)
The determinant is zero so we know that there is one root.
√
To check if the solution is correct, just verify that it satisfies the equation:
The solution x = 2 does indeed satisfy the equation.

Question 4.8
Find the roots of the equation given in Question 4.6
Answer
The equation given in question 4.6 is
The determinant of the equation, calculated above, is Δ = 25 and there are two
solutions.
We call the solutions
and
.
√
( )
√
So we get
and
Check that
and
.
satisfy the equation:
For
(
)
(
)
For
( )

( )
Question 4.9
Find the roots (if any) of the equation
Answer
First of all we put the equation in standard form, so that all non-zero terms
appear in one side of the equation:
Then we continue as before and calculate the discriminant:
a=1
b=
c=
(
)
( ) (
(
)
)
The determinant is positive, therefore there are two distinct roots
√
So we get
(
and
Check that
and
)
and
√
.
satisfy the equation:
For
0=0
For
(
)
(
0.0676 + 3.9 4 = 0
0=0

)
Question 4.10
Solve the equation
Give your answer in the form
Answer
Again, first re-arrange into standard form, then calculate the discriminant:
a=1
b=
c=
(
)
The determinant is positive, therefore there are two distinct roots.
We are asked to present the answer in the form
.
(
√
)
√
√
We can simplify the above expression by noting that
√
√
√
√
And dividing across by 2
√
So our final answer is
√
√